Modern Geometry— Methods and Applications: Part II: The Geometry and Topology of ManifoldsSpringer Science & Business Media, 05.08.1985 - 432 Seiten Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e. |
Inhalt
CHAPTER | 1 |
2 The simplest examples of manifolds | 10 |
3 Essential facts from the theory of Lie groups | 20 |
4 Complex manifolds | 31 |
5 The simplest homogeneous spaces | 41 |
CHAPTER 2 | 65 |
10 Various properties of smooth maps of manifolds | 77 |
11 Applications of Sards theorem | 90 |
22 Covering homotopies The homotopy groups of covering spaces | 193 |
23 Facts concerning the homotopy groups of spheres Framed normal | 207 |
Smooth Fibre Bundles | 220 |
25 The differential geometry of fibre bundles | 251 |
26 Knots and links Braids | 286 |
CHAPTER 7 | 297 |
28 Hamiltonian systems on manifolds Liouvilles theorem Examples | 308 |
29 Foliations | 322 |
CHAPTER 3 | 99 |
14 Applications of the degree of a mapping | 110 |
15 The intersection index and applications | 125 |
CHAPTER 4 | 135 |
18 Covering maps and covering homotopies | 148 |
19 Covering maps and the fundamental group Computation of | 157 |
CHAPTER 5 | 185 |
30 Variational problems involving higher derivatives | 333 |
CHAPTER 8 | 358 |
32 Some examples of global solutions of the YangMills equations | 393 |
33 The minimality of complex submanifolds | 414 |
423 | |
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Modern Geometry— Methods and Applications: Part II: The Geometry and ... B.A. Dubrovin,A.T. Fomenko,S.P. Novikov Keine Leseprobe verfügbar - 1984 |
Modern Geometry— Methods and Applications: Part II: The Geometry and ... B.A. Dubrovin,A.T. Fomenko,S.P. Novikov Keine Leseprobe verfügbar - 1984 |
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